Searching and Sorting Linear Search Binary Search Reading

Linear Search • Searching is the process of determining whether or not a given value exists in a data structure or a storage media. • We discuss two searching methods on one-dimensional arrays: linear search and binary search. • The linear (or sequential) search algorithm on an array is: – Sequentially scan the array, comparing each array item with the searched value. – If a match is found; return the index of the matched element; otherwise return – 1. • Note: linear search can be applied to both sorted and unsorted arrays. © 2006 Pearson Addison-Wesley. All rights reserved 2

Linear Search • The algorithm translates to the following Java method: public static int linear. Search(Object[] array, Object key) { for(int k = 0; k < array. length; k++) if(array[k]. equals(key)) return k; return -1; } © 2006 Pearson Addison-Wesley. All rights reserved 3

Binary Search • Binary search uses a recursive method to search an array to find a specified value • The array must be a sorted array: a[0]≤a[1]≤a[2]≤. . . ≤ a[final. Index] • If the value is found, its index is returned • If the value is not found, -1 is returned • Note: Each execution of the recursive method reduces the search space by about a half © 2006 Pearson Addison-Wesley. All rights reserved 4

Binary Search • An algorithm to solve this task looks at the middle of the array or array segment first • If the value looked for is smaller than the value in the middle of the array – Then the second half of the array or array segment can be ignored – This strategy is then applied to the first half of the array or array segment © 2006 Pearson Addison-Wesley. All rights reserved 5

Binary Search • If the value looked for is larger than the value in the middle of the array or array segment – Then the first half of the array or array segment can be ignored – This strategy is then applied to the second half of the array or array segment • If the value looked for is at the middle of the array or array segment, then it has been found • If the entire array (or array segment) has been searched in this way without finding the value, then it is not in the array © 2006 Pearson Addison-Wesley. All rights reserved 6

Pseudocode for Binary Search © 2006 Pearson Addison-Wesley. All rights reserved 7

Recursive Method for Binary Search © 2006 Pearson Addison-Wesley. All rights reserved 8

Execution of the Method search (Part 1 of 2) © 2006 Pearson Addison-Wesley. All

Checking the search Method 1. There is no infinite recursion • • On each recursive call, the value of first is increased, or the value of last is decreased If the chain of recursive calls does not end in some other way, then eventually the method will be called with first larger than last © 2006 Pearson Addison-Wesley. All rights reserved 11

Checking the search Method 2. Each stopping case performs the correct action for that case • • If first > last, there are no array elements between a[first] and a[last], so key is not in this segment of the array, and result is correctly set to 1 If key == a[mid], result is correctly set to mid © 2006 Pearson Addison-Wesley. All rights reserved 12

Checking the search Method 3. For each of the cases that involve recursion, if all recursive calls perform their actions correctly, then the entire case performs correctly • • • If key < a[mid], then key must be one of the elements a[first] through a[mid-1], or it is not in the array The method should then search only those elements, which it does The recursive call is correct, therefore the entire action is correct © 2006 Pearson Addison-Wesley. All rights reserved 13

Checking the search Method • • • If key > a[mid], then key must be one of the elements a[mid+1] through a[last], or it is not in the array The method should then search only those elements, which it does The recursive call is correct, therefore the entire action is correct The method search passes all three tests: Therefore, it is a good recursive method definition © 2006 Pearson Addison-Wesley. All rights reserved 14

Efficiency of Binary Search • The binary search algorithm is extremely fast compared to an algorithm that tries all array elements in order – About half the array is eliminated from consideration right at the start – Then a quarter of the array, then an eighth of the array, and so forth © 2006 Pearson Addison-Wesley. All rights reserved 15

Efficiency of Binary Search • Given an array with 1, 000 elements, the binary search will only need to compare about 10 array elements to the key value, as compared to an average of 500 for a serial search algorithm • The binary search algorithm has a worst-case running time that is logarithmic: O(log n) – A serial search algorithm is linear: O(n) • If desired, the recursive version of the method search can be converted to an iterative version that will run more efficiently © 2006 Pearson Addison-Wesley. All rights reserved 16